TY - JOUR

T1 - Third-order derivative matrix of a skew ray with respect to the source ray vector at a flat boundary

AU - Lin, Psang Dain

N1 - Funding Information:
Ministry of Science and Technology, Taiwan (106-2221-E-006-091-MY3).
Publisher Copyright:
© 2020 Optical Society of America

PY - 2019/9

Y1 - 2019/9

N2 - Our group recently showed that the Seidel primary ray aberration coefficients of an axis-symmetrical system can be accurately determined using the third-order Taylor series expansion of a skew ray R¯m on an image plane. This finding inspires us to determine the third-order derivative matrix of R¯m with respect to the vector X¯0 of the source ray, i.e., ∂R¯ 3m/∂X¯ 30, under reflection/refraction at a flat boundary. Finite difference methods using the second-order derivative matrix, ∂R¯ 2m/∂X¯ 20, require multiple rays to compute ∂R¯ 3m/∂X¯ 30 and suffer from cumulative rounding and truncation errors. By contrast, the present method is based on differential geometry. Thus, it provides a greater inherent accuracy and requires the tracing of just one ray. The proposed method facilitates the analytical investigation of the primary aberrations of an axis-symmetrical system and can be easily extended to determine the higher-order derivative matrices required to explore higher-order ray aberration coefficients.

AB - Our group recently showed that the Seidel primary ray aberration coefficients of an axis-symmetrical system can be accurately determined using the third-order Taylor series expansion of a skew ray R¯m on an image plane. This finding inspires us to determine the third-order derivative matrix of R¯m with respect to the vector X¯0 of the source ray, i.e., ∂R¯ 3m/∂X¯ 30, under reflection/refraction at a flat boundary. Finite difference methods using the second-order derivative matrix, ∂R¯ 2m/∂X¯ 20, require multiple rays to compute ∂R¯ 3m/∂X¯ 30 and suffer from cumulative rounding and truncation errors. By contrast, the present method is based on differential geometry. Thus, it provides a greater inherent accuracy and requires the tracing of just one ray. The proposed method facilitates the analytical investigation of the primary aberrations of an axis-symmetrical system and can be easily extended to determine the higher-order derivative matrices required to explore higher-order ray aberration coefficients.

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U2 - 10.1364/JOSAA.399620

DO - 10.1364/JOSAA.399620

M3 - Article

C2 - 32902432

AN - SCOPUS:85090819143

SN - 1084-7529

VL - 37

SP - 1435

EP - 1441

JO - Journal of the Optical Society of America A: Optics and Image Science, and Vision

JF - Journal of the Optical Society of America A: Optics and Image Science, and Vision

IS - 9

ER -